where the thermal corrections \(\Pi _i(T)\)  are given in appendix .
The one-loop potential becomes gauge-dependent when thermal corrections of bosons' masses are introduced [1]}, [2]},
leading to gauge-dependent critical temperature and GW spectrum produced from phase transition [3]}, [4]}.
To focus on our topic, in this work we take the Landau gauge with a vanishing gauge-fixing parameter (\(\xi =0\) ) for the effective potential.
In section  we will scrutinize the gauge dependence issue. There we will show that our main conclusions
made in the following sections with the \(\xi =0\)  effective potential are not changed when compared with the analyses made by using a gauge-independent effective potential proposed in Ref. [5]}.
